AIR POLLUTION – MONITORING MODELLING AND HEALTH

Analytical Model for Air Pollution in the Atmospheric Boundary Layer
Analytical Model for Air Pollution in the Atmospheric Boundary Layer 17
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Concluding, analytical solutions of equations are of fundamental importance in
understanding and describing physical phenomena, since they might take into account
all the parameters of a problem, and investigate their influence. Moreover, when using
models, while they are rather sophisticated instruments that ultimately reflect the current
state of knowledge on turbulent transport in the atmosphere, the results they provide are
subject to a considerable margin of error. This is due to various factors, including in particular
the uncertainty of the intrinsic variability of the atmosphere. Models, in fact, provide values
expressed as an average, i.e., a mean value obtained by the repeated performance of many
experiments, while the measured concentrations are a single value of the sample to which the
ensemble average provided by models refer. This is a general characteristic of the theory of
atmospheric turbulence and is a consequence of the statistical approach used in attempting
to parametrise the chaotic character of the measured data. An analytical solution can be
useful in evaluating the performances of numerical models (that solve numerically the
advection-diffusion equation) that could compare their results, not only against experimental
data but, in an easier way, with the solution itself in order to check numerical errors without
the uncertainties presented above. Finally, the program of providing analytical solutions for
realistic physical problems, leads us to future problems with different closure hypothesis
considering full space-time dependence in the resulting dynamical equation, which we will
also approach by the proposed methodology.
6. Acknowledgements
The authors thank to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico)
for the partial financial support of this work.
7. References
Abate, J. & Valkó, P.P. (2004). Multi-precision Laplace transform inversion. Int. J. for Num.
Methods in Engineering, Vol. 60, page numbers (979-993).
Adomian, G. (1984). A New Approach to Nonlinear Partial Differential Equations. J. Math.
Anal. Appl., Vol. 102, page numbers (420-434).
Adomian, G. (1988). A Review of the Decomposition Method in Applied Mathematics. J. Math.
Anal. Appl., Vol. 135, page numbers (501-544) .
Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method, Kluwer,
Boston, MA. .
Blackadar, A.K. (1997). Turbulence and diffusion in the atmosphere: lectures in Environmental
Sciences, Springer-Verlag.
Bodmann, B.; Vilhena, M.T.; Ferreira, L.S. & Bardaji, J.B. (2010). An analytical solver for
the multi-group two dimensional neutron-diffusion equation by integral transform
techniques. Il Nuovo Cimento C, Vol. 33, page numbers (199-206).
Boichenko, V.A.; Leonov, G.A. & Reitmann, V. (2005). NDimension theory for ordinary equations,
Teubner, Stuttgart.
Buske, D.; Vilhena, M.T.; Moreira, D.M. & Tirabassi, T. (2007a). An analytical solution of the
advection-diffusion equation considering non-local turbulence closure. Environ. Fluid
Mechanics, Vol. 7, page numbers (43-54).